Sine and Cosine Word Problems Quiz

Word Problems Involving Sine or Cosine Functions

• A periodic function has the form y(t) = a sin(bt + c) + d, where t ≥ 0.
• The mean level of a periodic function is the average value of the function over one period.
• The amplitude of a periodic function is the maximum deviation from the mean level.
• The period of a periodic function is the time taken to complete one cycle.

Finding the Function

• If the mean level is 10, amplitude is 5, period is π, and phase constant is 3, the function is y(t) = 5 sin(2t + 3) + 10.
• To find the maximum value of y, find the maximum value of the sine function (which is 1) and multiply it by the amplitude, then add the mean level.

Particle Motion

• The equation y = 4 sin(4πt - 6) + 7 represents an oscillatory motion.
• The period of this motion is 1/2.
• The mean level of this motion is 7.
• The phase constant is -6.
• The amplitude is 4.
• The angular velocity is 4π.

Velocity of a Particle

• The velocity of a particle is given by the equation v(t) = 3 sin(2πt - π) + 1.
• The maximum speed is 4.
• The minimum speed is 0.
• The velocity at t = 6 is 2.

Intensity of Sound

• The intensity of sound is given by the equation I = 5 cos(2πt/3 - π/2) + 55 dB.
• The amplitude of this sound is 5 dB.
• The maximum intensity heard is 60 dB.
• The minimum intensity heard is 50 dB.

Affection of a Boy

• The intensity of a boy's affection is modeled as L = 10 cos(πt/2 + π/2) + 5, where t is in hours.
• The intensity of his affection when he meets his girl at t = 1 is approximately 2.99.
• The intensity of his affection at t = 4 is approximately 15.